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Convergence of Random Batch Method with replacement for interacting particle systems

Zhenhao Cai, Jian-Guo Liu, Yuliang Wang

TL;DR

This work proves the convergence of the Random Batch Method with replacement (RBM-r), a kinetic Monte Carlo approach for first-order interacting particle systems, to the original IPS in Wasserstein-2 distance. Under assumptions of strong convexity and Lipschitz continuity for the drift and interaction kernels, the authors establish an explicit convergence rate $W_2 \lesssim \sqrt{1+T}\,\kappa^{1/4}$ for the first marginal, with an improved rate when diffusion vanishes. The analysis hinges on a sophisticated coupling that introduces an intermediate IPS' dynamics and leverages a random-time-change together with a diagonal bridge to compare trajectories and distributions. The results justify RBM-r as a scalable, accurate method for simulating IPS and point to potential applications to other kinetic Monte Carlo schemes, including the stochastic Ising model, under appropriate conditions.

Abstract

The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is an efficient algorithm for simulating interacting particle systems (IPS). In this paper, we investigate the Random Batch Method with replacement (RBM-r), which is the same as the kinetic Monte Carlo (KMC) method for the pairwise interacting particle system of size $N$. In the RBM-r algorithm, one randomly picks a small batch of size $p \ll N$, and only the particles in the picked batch interact among each other within the batch for a short time, where the weak interaction (of strength $\frac{1}{N-1}$) in the original system is replaced by a strong interaction (of strength $\frac{1}{p-1}$). Then one repeats this pick-interact process. This KMC algorithm dramatically reduces the computational cost from $O(N^2)$ to $O(pN)$ per time step, and provides an unbiased approximation of the original force/velocity field of the interacting particle system. We give a rigorous proof of this approximation with an explicit convergence rate. In detail, we show that the Wasserstein-2 distance between first marginal distributions of IPS and RBM-r has an $O(κ^{1/4})$ upper bound, where $κ$ is the time step for choosing the random batch and the bound is independent of $N$. An improved $O(κ^{1/2})$ rate is also obtained when there is no diffusion in the system. Notably, the techniques in our analysis can potentially be applied to study KMC for other systems, including the stochastic Ising spin system.

Convergence of Random Batch Method with replacement for interacting particle systems

TL;DR

This work proves the convergence of the Random Batch Method with replacement (RBM-r), a kinetic Monte Carlo approach for first-order interacting particle systems, to the original IPS in Wasserstein-2 distance. Under assumptions of strong convexity and Lipschitz continuity for the drift and interaction kernels, the authors establish an explicit convergence rate for the first marginal, with an improved rate when diffusion vanishes. The analysis hinges on a sophisticated coupling that introduces an intermediate IPS' dynamics and leverages a random-time-change together with a diagonal bridge to compare trajectories and distributions. The results justify RBM-r as a scalable, accurate method for simulating IPS and point to potential applications to other kinetic Monte Carlo schemes, including the stochastic Ising model, under appropriate conditions.

Abstract

The Random Batch Method (RBM) proposed in [Jin et al. J Comput Phys, 2020] is an efficient algorithm for simulating interacting particle systems (IPS). In this paper, we investigate the Random Batch Method with replacement (RBM-r), which is the same as the kinetic Monte Carlo (KMC) method for the pairwise interacting particle system of size . In the RBM-r algorithm, one randomly picks a small batch of size , and only the particles in the picked batch interact among each other within the batch for a short time, where the weak interaction (of strength ) in the original system is replaced by a strong interaction (of strength ). Then one repeats this pick-interact process. This KMC algorithm dramatically reduces the computational cost from to per time step, and provides an unbiased approximation of the original force/velocity field of the interacting particle system. We give a rigorous proof of this approximation with an explicit convergence rate. In detail, we show that the Wasserstein-2 distance between first marginal distributions of IPS and RBM-r has an upper bound, where is the time step for choosing the random batch and the bound is independent of . An improved rate is also obtained when there is no diffusion in the system. Notably, the techniques in our analysis can potentially be applied to study KMC for other systems, including the stochastic Ising spin system.
Paper Structure (12 sections, 8 theorems, 113 equations, 3 figures, 2 algorithms)

This paper contains 12 sections, 8 theorems, 113 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. A preview of the main theorem and key techniques
  3. Lemmas used in the proof of Theorem \ref{['eq:thm:mainthm']}
  4. Random Batch Method with replacement (RBM-r) and convergence analysis
  5. Assumptions and statement of the main result
  6. Proof of the main result
  7. Numerical example
  8. Conclusion and discussions
  9. Proofs of Lemmas in Section \ref{['sec:lemmas']}
  10. Proof of Lemma \ref{['lmm:moment']}
  11. Proof of Lemma \ref{['lmm:hatX_continuity']}
  12. Proof of Lemma \ref{['eq:L2LLN']}

Key Result

Theorem 1

Consider the interacting particle system eq:IPSintro0 and the RBM-r algorithm eq:RBMintro. Assume the following conditions: Then, fixing $T>0$, for any small time step $\kappa$, the Wasserstein-2 distance between first marginal distributions of eq:IPSintro0 and eq:RBMintro is bounded by where $C$ is a positive constant only depending on $\lambda$, $L$, $T$, and $\bar{t}:= \tfrac{N}{p}t$ denotes

Figures (3)

  • Figure 1: Evolution of particle positions of RBM-r dynamics and original dynamics
  • Figure 2: Evolution of particle marginal distributions of RBM-r dynamics and original dynamics
  • Figure 3: Evolution of particle distributions of RBM-r dynamics and original dynamics

Theorems & Definitions (16)

  • Theorem
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3: $L^2$ weak law of large number
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Theorem 3.1
  • Remark 3.1: Discussion on different running times
  • ...and 6 more